An ultraweak variational method for parameterized linear differential-algebraic equations

TL;DR

This paper introduces an ultraweak variational approach for parameterized linear differential-algebraic equations, enabling stable discretization, model reduction, and error estimation, with demonstrated efficiency in numerical experiments.

Contribution

It develops a novel ultraweak variational formulation for DAEs, integrating it with Petrov-Galerkin and Reduced Basis methods for stable, efficient approximation and model reduction.

Findings

Significant reduction in system size achieved.

Stable and certified discretization method developed.

Effective combination with Balanced Truncation demonstrated.

Abstract

We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations (DAEs) w.r.t. the time variable which yields an optimally stable system. This is used within a Petrov-Galerkin method to derive a certified detailed discretization which provides an approximate solution in an ultraweak setting as well as for model reduction w.r.t. time in the spirit of the Reduced Basis Method (RBM). A computable sharp error bound is derived. Numerical experiments are presented that show that this method yields a significant reduction and can be combined with well-known system theoretic methods such as Balanced Truncation to reduce the size of the DAE.